Parametric Curve(s)
$$$ \forall\alpha\in\{0,2\pi\}\quad0.9\leqslant\;k_{a}\;\leqslant0.12$$$
$$$
\left \{ \begin{array}{r c l}
\theta(\alpha)& = &\alpha \\
r(\alpha)& = &k_{a}\times\left(1+cos^2{(2\cdot\alpha)}\right)
\end{array}\right .$$$
$$$\theta(\alpha)$$$